#6608919

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1463 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,7)(4,8)(5,6)(9,10)(11,16)(12,14)(13,15)(17,18,22,21)(19,23,27,25)(20,28,30,32)(24,29,31,26);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19,22,27)(18,23,21,25)(20,29,30,26)(24,28,31,32);(1,3,5,4)(2,7,6,8)(10,15)(11,12)(17,20,27,29,22,30,19,26)(18,24,23,32,21,31,25,28);(1,4,5,3)(2,8,6,7)(9,12,13,11)(10,14,15,16)(17,21,22,18)(19,23,27,25)(20,24,30,31)(26,28,29,32);(11,12)(14,16)(17,19,22,27)(18,23,21,25)(20,26,30,29)(24,32,31,28)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=fgi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6516115);